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Isomorphisms of Symplectic Torus Quotients

Hans-Christian Herbig, Gerald W. Schwarz, Christopher Seaton

Abstract

We call a reductive complex group $G$ quasi-toral if $G^0$ is a torus. Let $G$ be quasi-toral and let $V$ be a faithful $1$-modular $G$-module. Let $N$ (the shell) be the zero fiber of the canonical moment mapping $μ\colon V\oplus V^*\to\mathfrak{g}^*$. Then $N$ is a complete intersection variety with rational singularities. Let $M$ denote the categorical quotient $N/\!\!/ G$. We show that $M$ determines $V\oplus V^*$ and $G$, up to isomorphism, if $\operatorname{codim}_N N_\mathrm{sing}\geq 4$. If $\operatorname{codim}_NN_\mathrm{sing}=3$, the lowest possible, then there is a process to produce an algebraic (hence quasi-toral) subgroup $G'\subset G$ and a faithful $1$-modular $G'$-submodule $V'\subset V$ with shell $N'$ such that $\operatorname{codim}_{N'}(N')_\mathrm{sing}\geq 4$. Moreover, there is a $G'$-equivariant morphism $N'\to N$ inducing an isomorphism $N'/\!\!/ G'\xrightarrow{\sim} N/\!\!/ G$. Thus, up to isomorphism, $M$ determines $V'\oplus (V')^*$ and $G'$, hence also $N'$. We establish similar results for real shells and real symplectic quotients associated to unitary modules for compact Lie groups.

Isomorphisms of Symplectic Torus Quotients

Abstract

We call a reductive complex group quasi-toral if is a torus. Let be quasi-toral and let be a faithful -modular -module. Let (the shell) be the zero fiber of the canonical moment mapping . Then is a complete intersection variety with rational singularities. Let denote the categorical quotient . We show that determines and , up to isomorphism, if . If , the lowest possible, then there is a process to produce an algebraic (hence quasi-toral) subgroup and a faithful -modular -submodule with shell such that . Moreover, there is a -equivariant morphism inducing an isomorphism . Thus, up to isomorphism, determines and , hence also . We establish similar results for real shells and real symplectic quotients associated to unitary modules for compact Lie groups.
Paper Structure (8 sections, 53 theorems, 37 equations)

This paper contains 8 sections, 53 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Stable actions and symplectic slices
  4. Codimension of singularities and slice representations of type O
  5. Determining $\dim G$ topologically
  6. Quotienting by slice representations of type O
  7. Lifting Isomorphisms
  8. Consequences for Real Symplectic Quotients

Key Result

Theorem 1

Let $G$ be a torus and $V$ a $1$-modular faithful $G$-module. Let $H$ be the isotropy group of a nonzero closed orbit in $N$. Then, UTCLS, $V$ is stable, $V^H$ is a stable $G$-module and $H$ is the principal isotropy group of $V^H$.

Theorems & Definitions (98)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Theorem 8
  • Lemma 2.1
  • ...and 88 more